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Products and Factors of Polynomials
Objective: To multiply polynomials; to divide polynomials by long division
and synthetic division.
)52(3 xx )4)(7( xx )8)(8( xx
)52(3 xx )4)(7( xx
2
2
)1(4
)12(4
x
xx )32)(23( xx
)12)(12)(14(
)14)(14(2
22
xxx
xx
)8)(8( xx
Multiplying Polynomials
• When multiplying two binomials, you FOIL. However, what we are really doing is the distributive property. When multiplying trinomials or larger, all we can do is distribute.
Example 1
• Multiply the following: A B C
)132)(3( 23 xxx)2524(3 232 xxxx )1)(2)(3( xxx
Example 1
• Multiply the following: A B C
)132)(3( 23 xxx)2524(3 232 xxxx )1)(2)(3( xxx
2345 615612 xxxx
Example 1
• Multiply the following: A B C
)132)(3( 23 xxx)2524(3 232 xxxx )1)(2)(3( xxx
2345 615612 xxxx xxx 34 32396 23 xx
3992 234 xxxx
Example 1
• Multiply the following: A B C
)132)(3( 23 xxx)2524(3 232 xxxx )1)(2)(3( xxx
2345 615612 xxxx )23)(3( 2 xxx
xxx 23 23
693 2 xx
673 xx
xxx 34 32396 23 xx
3992 234 xxxx
Example 2
Example 2
Example 2
Try This
• Factor the following polynomials.
2223 xxxxx 93
Try This
• Factor the following polynomials.
2223 xxxxx 93
)3)(3(
)9( 2
xxx
xx
Try This
• Factor the following polynomials.
2223 xxxxx 93
)3)(3(
)9( 2
xxx
xx
)2)(1(
)1(2)1(2
2
xx
xxx
Sum/Difference of 2 Cubes
• To factor the sum or difference of 2 cubes, you must memorize the following definitions.
))(( 2233 babababa
))(( 2233 babababa
Example 3
Example 3
Example 3
Try This
• Factor the following polynomials.
10003 x 1253 x
Try This
• Factor the following polynomials.
10003 x 1253 x
)10010)(10( 2 xxx
Try This
• Factor the following polynomials.
10003 x 1253 x
)10010)(10( 2 xxx )255)(5( 2 xxx
The Factor Theorem
• The factor theorem states the relationship between the linear factors of a polynomial expression and the terms of the related polynomial functions
• (x – r) is a factor of the polynomial expression that defines the function P if and only if r is a solution of P(x) = 0, that is, if and only if P(r) = 0.
Example 4
Example 4
Example 4
Try This
• Use substitution to determine whether x + 3 is a factor of x3 – 3x2 -6x + 8.
Try This
• Use substitution to determine whether x + 3 is a factor of x3 – 3x2 -6x + 8.
• x + 3 needs to be rewritten as x – (-3), so r = -3.
solution anot ;0)3(
288182727)3(
8)3(6)3(3)3()3( 23
P
P
P
Dividing Polynomials
• A polynomial can be divided by a divisor of the form x – r by using long division or by a method called synthetic division. Long division of polynomials is similar to long division of real numbers. We will follow the same pattern.
Dividing Polynomials
• Divide the following:65
12432
23
xx
xxx
Dividing Polynomials
• Divide the following:
• What do we multiply x2 by to get x3 ? x.
65
12432
23
xx
xxx
124365 232 xxxxx
Dividing Polynomials
• Divide the following:
• What do we multiply x2 by to get x3 ? x.• Subtract.
65
12432
23
xx
xxx
124365 232 xxxxx
x
)65( 23 xxx
xx 102 2
Dividing Polynomials
• Divide the following:
• What do we multiply x2 by to get x3 ? x.• Subtract.• Bring down the -12.
65
12432
23
xx
xxx
124365 232 xxxxx
x
)65( 23 xxx
12102 2 xx
Dividing Polynomials
• Divide the following:
• What do we multiply x2 by to get x3 ? x.• Subtract.• Bring down the -12.• What do we multiply x2 by to get -2x2 ? -2.• Subtract.
65
12432
23
xx
xxx
124365 232 xxxxx
2x
)65( 23 xxx
12102 2 xx
)12102( 2 xx
0
Example 5
Example 5
Try This
• Find the quotient. )32()15133( 223 xxxxx
Try This
• Find the quotient. )32()15133( 223 xxxxx
1513332 232 xxxxx
5x
)32( 23 xxx
15105 2 xx
)15105( 2 xx
0
Synthetic Division
• With synthetic division, we will only write the coefficients, not the variables. Again, you will need to memorize the following problem solving skill.
Synthetic Division
• Divide the following using synthetic division.
2
1243 23
x
xxx
Synthetic Division
• Divide the following using synthetic division.
2 | 1 3 -4 -12
2
1243 23
x
xxx
Synthetic Division
• Divide the following using synthetic division.
2 | 1 3 -4 -12
1
2
1243 23
x
xxx
Synthetic Division
• Divide the following using synthetic division.
2 | 1 3 -4 -12 2 1 5
2
1243 23
x
xxx
Synthetic Division
• Divide the following using synthetic division.
2 | 1 3 -4 -12 2 10 1 5 6
2
1243 23
x
xxx
Synthetic Division
• Divide the following using synthetic division.
2 | 1 3 -4 -12 2 10 12 1 5 6 0
2
1243 23
x
xxx
Synthetic Division
• Divide the following using synthetic division.
2 | 1 3 -4 -12 2 10 12 1 5 6 0• The answer is
2
1243 23
x
xxx
652 xx
Try This
• Divide the following using synthetic division.
3
832 23
x
xxx
Try This
• Divide the following using synthetic division.
-3 | 1 2 3 8 -3 3 -18 1 -1 6 -10• The answer is
3
832 23
x
xxx
10
62
r
xx
Remainder
• If there is a nonzero remainder after dividing with either method, it is usually written as the numerator of a fraction, with the divisor as the denominator.
• Also notice how every power of x needs to be there.• Divide
)3()48( 3 xx
Remainder
• If there is a nonzero remainder after dividing with either method, it is usually written as the numerator of a fraction, with the divisor as the denominator.
• Also notice how every power of x needs to be there.• Divide
-3| 1 0 0 48
)3()48( 3 xx
))3(()4800( 23 xxxx
Remainder
• If there is a nonzero remainder after dividing with either method, it is usually written as the numerator of a fraction, with the divisor as the denominator.
• Also notice how every power of x needs to be there.• Divide
-3| 1 0 0 48 -3 9 -27 1 -3 9 21
)3()48( 3 xx
3212 93 xxx
))3(()4800( 23 xxxx
Example 6
Example 6
Example 7
Example 7
Try This
• Given )3( find ,4323)( 23 PxxxxP
Try This
• Given
3| 3 2 -3 4 9 33 90 3 11 30 94
)3( find ,4323)( 23 PxxxxP
Try This
• Given
3| 3 2 -3 4 9 33 90 3 11 30 94
)3( find ,4323)( 23 PxxxxP
94491881)3(
4)3(3)3(2)3(3)3( 23
P
P
8262824426 23223 xxxxxxxx
)2)(5(3)103(3 2 xxxxxx )497)(7( 2 xxx
8262824426 23223 xxxxxxxx
)2)(5(3)103(3 2 xxxxxx )497)(7( 2 xxx
noP ;3516)3(4)3(4)3()3( 23 yesP ;016)4(4)4(4)4()4( 23
4x
)4)(3)(2( xxx
Homework
• Pages 445-446• 15-96 multiples of 3• Skip 33
